Neutrino Mass Spectrum, Majorana CP-Violation, $\beta\beta$-Decay and Beyond

The problem of determination of the nature - Dirac or Majorana, of massive neutrinos is discussed. The physics potential of experiments, searching for $\beta\beta-$decay, for providing information on the type of $\nu$-mass spectrum, absolute scale of $\nu$- masses and on the Majorana phases in the PMNS neutrino mixing matrix $U$, is reviewed. The possibility that the CP-violation necessary for the generation of the baryon asymmetry of the Universe is due exclusively to the Majorana CP-violating phase(s) in the PMNS neutrino mixing matrix $U$, is also briefly discussed.